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TS Inter 2nd Year Maths 2B Question Paper May 2022
Time : 3 Hours
Max. Marks : 75
Section – A
(10 × 2 = 20)
1. Find the equation of the circle passing through (2, -1) having the centre at (2, 3).
2. If x2 + y2 – 4x + 6y + c = 0 represents a circle with radius 6, then find the value of c.
3. Find the value of k if the points (1, 3) and (2, k) are conjugate with respect to the circle x2 + y2 = 35.
4. Find the chord of contact (0, 5) with respect to the circle x2 + y2 – 5x + 4y – 2 = 0.
5. Find the angle between the circles x2 + y2 + 6x – 10y – 135 = 0 and x2 + y2 – 4x + 14y – 116 = 0.
6. Find the common tangent of the circles x2 + y2 + 10x – 2y + 22 = 0 and x2 + y2 + 2x – 8y + 8 = 0 at their point of contact.
7. Find the equation of the parabola whose vertex is (3, -2) and focus is (3, 1).
8. If the eccentricity of a hyperbola is \(\frac{5}{4}\), then find the eccentricity of its conjugate hyperbola.
9. Evaluate \(\int\left[\frac{1}{1-x^2}+\frac{1}{1+x^2}\right]\) dx on (-1, 1).
10. Evaluate ∫(x3 – 2x2 + 3) dx on R.
11. Evaluate ∫\(\frac{\mathrm{e}^{\tan -1}}{1+\mathrm{x}^2}\)dx in I ⊂ (0, ∞)
12. Evaluate \(\int \frac{3 x^2}{1+x^6}\) dx on R.
13. Evaluate \(\int_0^5(x+1) d x .\)
14. Evaluate \(\int_0^\pi \sqrt{2+2 \cos \theta}\) dθ.
15. Find the order and degree of the differential equation
\(\frac{\mathrm{d}^2 \mathrm{y}}{\mathrm{dx}}\) = \(\left[1+\left(\frac{d y}{d x}\right) 2\right]^{\frac{5}{3}} \text {. }\)
Section – B
II. Short Answer Type Questions.
- Answer any FIVE questions.
- Each Question carries FOUR marks.
16. If the length of the tangent from (2, 5) to the circle x2 + y2 – 5x + 4y + k = 0 is \(\sqrt{37}\) then find k.
17. Find the pole of 3x + 4y – 45 = 0 with respect to x2 + y2 – 6x – 8y + 5 = 0
18. Find the angle between the tangents drawn from (3, 2) to the circle x2 + y2 – 4x + 2y – 7 = 0.
19. Find the equation of the circle which cuts orthogonally the circle x2 + y2 – 4x + 2y – 7 = 0 and having the centre at (2, 3).
20. Show that the circles x2 + y2 – 8x – 2y + 8 = 0 and x2 + y2 – 2x + 6y + 6 = 0 touch each other and find the point of contact.
21. Find the equation of ellipse in the standard form if it passes through the points (-2, 2) and (3, -1).
22. Find the equation of ellipse in the standard from whose distance from foci is 2 and the length of latus rectum is \(\frac{15}{2}\).
23. Find the equation of the hyperbola whose foci are (± 5, 0), the transverse axis is of length 8.
24. Evaluate \(\int_0^2|1-x|\)| dx.
25. Evaluate \(\int_{-1}^2 \frac{x^2}{x^2+2}\) dx.
26. Solve the differential equation \(\frac{d y}{d x}\) = \(\frac{x y+y}{x y+x}\)
27. Solve the differential equation \(\frac{d y}{d x}\) = ex-y + x2e-y.
Section – C
(5 × 7 = 35)
III. Long Answer Type Questions.
- Answer ANY FIVE questions.
- Each Question carries SEVEN marks.
28. Find the equation of the circle passing through the points (3, 4), (3, 2) and (1, 4).
29. Find the length of the chord intercepted by the circle x2 + y2 – x + 3y – 22 = 0 on the line y = x – 3.
30. Find the equation of the circle which touches the circle x2 + y2 – 2x – 4y – 20 = 0 externally at (5, 5) with radius 5.
31. Find the equation of the circle passing through origin, having its centre on the line x + y = 4 and intersecting the circle x2 + y2 – 4x + 2y + 4 = 0
32. Derive the equation of the parabola in standard form.
33. Evaluate \(\int \frac{\left(a^x-b^x\right)^2}{a^x b^x}\) dx, (a > 0, a ≠ 1, b > 0, b ≠ 1) on R.
34. Evaluate \(\int \frac{1}{(x+3) \sqrt{x+2}}\) dx on I ⊂ (-2, ∞).
35. Evaluate \(\int \frac{d x}{\cos ^2 x+\sin 2 x}\)
I ⊂ R – |(2n + 1)\(\frac{\pi}{2}\), n ∈ z} ∪ {2nπ + tan-1\(\frac{1}{2}\), n ∈ z}.
36. Evaluate \(\int_0^{\frac{\pi}{4}}\)log(1 + tan x) dx.
37. Solve the differential equation \(\frac{\mathrm{dy}}{\mathrm{dx}}\) = tan2 (x + y)